Biswas, SurajitSurajitBiswasSaurabh, BipulBipulSaurabh2025-08-282025-08-282025-04-0110.48550/arXiv.2505.01439https://d8.irins.org/handle/IITG2025/20179We compute the spectral dimension, the dimension of a symmetric random walk, and the Gelfand-Kirillov dimension for compact Vilenkin groups. As a result, we show that these dimensions are zero for any compact, totally disconnected, metrizable topological group. We provide an explicit description of the -groups for compact Vilenkin groups. We express the generators of the -groups in terms of the corresponding matrix coefficients for two specific examples: the group of -adic integers and the -adic Heisenberg group. Finally, we prove the nonexistence of a natural class of spectral triples on the group of -adic integers.en-USCompact Vilenkin groupSpectral dimensionRandom walkK-groupsSpectral triplese-Printe-Print123456789/555